Riemann Hypothesis and Random Walks: the Zeta Case
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Riemann Hypothesis and Random Walks: the Zeta case Andr´e LeClaira Cornell University, Physics Department, Ithaca, NY 14850 Abstract In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for 1 its L-function is valid to the right of the critical line <(s) > 2 , and the Riemann Hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a 1-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. arXiv:1601.00914v3 [math.NT] 23 May 2017 a [email protected] 1 I. INTRODUCTION There are many generalizations of Riemann's zeta function to other Dirichlet series, which are also believed to satisfy a Riemann Hypothesis. A common opinion, based largely on counterexamples, is that the L-functions for which the Riemann Hypothesis is true enjoy both an Euler product formula and a functional equation. However a direct connection between these properties and the Riemann Hypothesis has not been formulated in a precise manner. In [1, 2] a concrete proposal making such a connection was presented for Dirichlet L-functions, and those based on cusp forms, due to the validity of the Euler product formula to the right of the critical line. In contrast to the non-principal case, in this approach the case of principal Dirichlet L-functions, of which Riemann zeta is the simplest, turned out to be more delicate, and consequently it was more difficult to state precise results. In the present work we address further this special case. Let χ(n) be a Dirichlet character modulo k and L(s; χ) its L-function with s = σ + it. It satisfies the Euler product formula 1 1 −1 X χ(n) Y χ(pn) L(s; χ) = = 1 − (1) ns ps n=1 n=1 n where pn is the n-th prime. The above formula is valid for <(s) > 1 since both sides converge absolutely. The important distinction between principal verses non-principal characters is the following. For non-principal characters the L-function has no pole at s = 1, thus there exists the possibility that the Euler product is valid partway inside the strip, i.e. has 1 abscissa of convergence σc < 1. It was proposed in [1, 2] that σc = 2 for this case. In contrast, now consider L-functions based on principal characters. The latter character is defined as χ(n) = 1 if n is coprime to k and zero otherwise. The Riemann zeta function is the trivial principal character of modulus k = 1 with all χ(n) = 1. L-functions based on principal characters do have a pole at s = 1, and therefore have abscissa of convergence σc = 1, which implies the Euler product in the form given above cannot be valid inside the critical strip 0 < σ < 1. Nevertheless, in this paper we will show how a truncated version of 1 the Euler product formula is valid for σ > 2 . The primary aim of the work [1, 2] was to determine what specific properties of the prime numbers would imply that the Riemann Hypothesis is true. This is the opposite of the more well-studied question of what the validity of the Riemann Hypothesis implies 2 for the fluctuations in the distribution of primes. The answer proposed was simply based on the multiplicative independence of the primes, which to a large extent underlies their pseudo-random behavior. To be more specific, let χ(n) = eiθn for χ(n) 6= 0. In [1, 2] it was proven that if the series N X BN (t; χ) = cos (t log pn + θpn ) (2) n=1 p is O( N), then the Euler product converges for σ > 1 and the formula (1) is valid to the 2 p right of the critical line. In fact, we only need BN = O( N) up to logs (see Remark 1); p when we write write O( N), it is implicit that this can be relaxed with logarithmic factors. For non-principal characters the allowed angles θn are equally spaced on the unit circle, and it was conjectured in [2] that the above series with t = 0 behaves like a random walk due to p the multiplicative independence of the primes, and this is the origin of the O( N) growth. Furthermore, this result extends to all t since domains of convergence of Dirichlet series are always half-planes. Taking the logarithm of (1), one sees that log L is never infinite to the right of the critical line and thus has no zeros there. This, combined with the functional equation that relates L(s) to L(1−s), implies there are also no zeros to the left of the critical line, so that all zeros are on the line. The same reasoning applies to cusp forms if one also uses a non-trivial result of Deligne [2]. In this article we reconsider the principal Dirichlet case, specializing to Riemann zeta itself since identical arguments apply to all other principal cases with k > 1. Here all angles θn = 0, so one needs to consider the series N X BN (t) = cos(t log pn) (3) n=1 which now strongly depends on t. On the one hand, whereas the case of principal Dirichlet L- functions is complicated by the existence of the pole, and, as we will see, one consequently needs to truncate the Euler product to make sense of it, on the other hand BN can be estimated using the prime number theorem since it does not involve sums over non-trivial characters χ, and this aids the analysis. This is in contrast to the non-principal case, where, however well-motivated, we had to conjecture the random walk behavior alluded to above, so in this respect the principal case is potentially simpler. To this end, a theorem of Kac p (Theorem 1 below) nearly does the job: BN (t) = O( N) in the limit t ! 1, which is also a consequence of the multiplicative independence of the primes. This suggests that one can 3 also make sense of the Euler product formula in the limit t ! 1. However this is not enough for our main purpose, which is to have a similar result for finite t which we will develop. This article is mainly based on our previous work [1, 2] but provides a more detailed analysis and extends it in several ways. It was suggested in [1] that one should truncate the series at an N that depends on t. First, in the next section we explain how a simple group structure underlies a finite Euler product which relates it to a generalized Dirichlet series which is a subseries of the Riemann zeta function. Subsequently we estimate the error under truncation, which shows explicitly how this error is related to the pole at s = 1, as expected. The remainder of the paper, sections IV-VI, presents various applications of these ideas. We use them to study the argument of the zeta function. We present an algorithm to calculate very high zeros, far beyond what is currently known. We also study the statistical fluctuations of individual zeros, in other words, a 1-point correlation function. In many respects, our work is related to the work of Gonek et. al. [4, 5], which also considers a truncated Euler product. The important difference is that the starting point in [4] is a hybrid version of the Euler product which involves both primes and zeros of zeta. Only after assuming the Riemann Hypothesis can one explain in that approach why the truncated product over primes is a good approximation to zeta. In contrast, here we do not assume anything about the zeros of zeta, since the goal is to actually understand their location. We are unable to provide fully rigorous proofs of some of the statements below, however we do provide supporting calculations and numerical work. In order to be clear on this, below \Proposal" signifies the most important claims that we could not rigorously prove. II. ALGEBRAIC STRUCTURE OF FINITE EULER PRODUCTS The aim of this section is to define properly the objects we will be dealing with. In par- ticular we will place finite Euler products on the same footing as other generalized Dirichlet series. The results are straightforward and are mainly definitions. Definition 1. Fix a positive integer N and let fp1; p2; : : : pN g denote the first N primes where p1 = 2. From this set one can generate an abelian group QN of rank N with elements n o n1 n2 nN QN = p1 p2 ··· pN ; ni 2 Z 8i (4) 4 + + where the group operation is ordinary multiplication. Clearly QN ⊂ Q where Q are the positive rational numbers. There are an infinite number of integers in QN which form a subset of the natural numbers N = f1; 2;:::g. We will denote this set as NN ⊂ N, and elements of this set simply as n. Definition 2. Fix a positive integer N. For every integer n 2 N we can define the character c(n): c(n) = 1 if n 2 NN ⊂ QN = 0 otherwise (5) Clearly, for a prime p, c(p) = 0 if p > pN .